The angle $B$ of a triangle $ABC$ is $120^o$. Let $M$ be a point on the side $AC$ and $K$ a point on the extension of the side $AB$, such that $BM$ is the internal bisector of the angle $\angle ABC$ and $CK$ is the external bisector corresponding to the angle $\angle ACB$ . The segment $MK$ intersects $BC$ at point $P$. Prove that $\angle APM = 30^o$.

Let $ABC$ be a triangle. The points $K, L,$ and $M$ lie on the segments $BC, CA,$ and $AB,$ respectively, such that the lines $AK, BL,$ and $CM$ intersect in a common point. Prove that it is possible to choose two of the triangles $ALM, BMK,$ and $CKL$ whose inradii sum up to at least the inradius of the triangle $ABC$. [i]Proposed by Estonia[/i]

Let $A$ be a point inside the acute angle $xOy$. An arbitrary circle $\omega$ passes through $O, A$, intersecting $Ox$ and $Oy$ at the second intersection $B$ and $C$, respectively. Let $M$ be the midpoint of $BC$. Prove that $M$ is always on a fixed line (when $\omega$ changes, but always goes through $O$ and $A$).

Let there be an incircle of triangle $ABC$, and 3 circles each inscribed between incircle and angles of $ABC$. Let $r, r_1, r_2, r_3$ be radii of these circles ($r_1, r_2, r_3 < r$). Prove that $$r_1+r_2+r_3 \geq r$$

Let $\Gamma_1$ and $\Gamma_2$ be circles internally tangent at point $A$, with $\Gamma_1$ inside $\Gamma_2$. Let $BC$ be a chord of $\Gamma_2$ which is tangent to $\Gamma_1$ at point $D$. Prove that line $AD$ is the angle bisector of $\angle BAC$.

Let $ABC$ be an acute triangle. Let $D$, $E$ and $F$ be the feet of the altitudes from $A$, $B$ and $C$ respectively and let $H$ be the orthocenter of $\triangle ABC$. Let $X$ be an arbitrary point on the circumcircle of $\triangle DEF$ and let the circumcircles of $\triangle EHX$ and $\triangle FHX$ intersect the second time the lines $CF$ and $BE$ second at $Y$ and $Z$, respectively. Prove that the line $YZ$ passes through the midpoint of $BC$.

Given a tetrahedron $A_1A_2A_3A_4$, define an $A_1$-exsphere such a sphere that is tangent to all planes given by faces of the tetrahedron and the vertex $A_1$ and the sphere are separated by the plane $A_2A_3A_4.$ Denote $\varrho_1,\ldots,\varrho_4$ of all four exspheres. Furthermore, denote $v_i, i=1,\ldots,4$ the distance of the vertex $A_i$ from the opposite face. Show that \[2\left(\frac{1}{v_1}+\frac{1}{v_2}+\frac{1}{v_3}+\frac{1}{v_4}\right)=\frac{1}{\varrho_1}+\frac{1}{\varrho_2}+\frac{1}{\varrho_3}+\frac{1}{\varrho_4}.\]

A rectangular piece of paper $ABCD$ has sides of lengths $AB = 1$, $BC = 2$. The rectangle is folded in half such that $AD$ coincides with $BC$ and $EF$ is the folding line. Then fold the paper along a line $BM$ such that the corner $A$ falls on line $EF$. How large, in degrees, is $\angle ABM$? [asy] size(180); pathpen = rgb(0,0,0.6)+linewidth(1); pointpen = black+linewidth(3); pointfontpen = fontsize(10); pen dd = rgb(0,0,0.6) + linewidth(0.7) + linetype("4 4"), dr = rgb(0.8,0,0), dg = rgb(0,0.6,0), db = rgb(0,0,0.6)+linewidth(1); pair A=(0,1), B=(0,0), C=(2,0), D=(2,1), E=A/2, F=(2,.5), M=(1/3^.5,1), N=reflect(B,M)*A; D(B--M--D("N",N,NE)--B--D("C",C,SE)--D("D",D,NE)--M); D(D("M",M,plain.N)--D("A",A,NW)--D("B",B,SW),dd); D(D("E",E,W)--D("F",F,plain.E),dd); [/asy]

The vertices A,B,C of an acute-angled triangle ABC lie on the sides B1C1, C1A1, A1B1 respectively of a triangle A1B1C1 similar to the triangle ABC (∠A = ∠A1, etc.). Prove that the orthocenters of triangles ABC and A1B1C1 are equidistant from the circumcenter of △ABC.

Let $I_a$ be the point of the center of an ex-circle of the triangle $ABC$, which touches the side $BC$ . Let $W$ be the intersection point of the bisector of the angle $\angle A$ of the triangle $ABC$ with the circumcircle of the triangle $ABC$. Perpendicular from the point $W$ on the straight line $AB$, intersects the circumcircle of $ABC$ at the point $P$. Prove, that if the points $B, P, I_a$ lie on the same line, then the triangle $ABC$ is isosceles. (Mykola Moroz)

In $\triangle ABC$ shown in the figure, $AB=7$, $BC=8$, $CA=9$, and $\overline{AH}$ is an altitude. Points $D$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $\overline{BD}$ and $\overline{CE}$ are angle bisectors, intersecting $\overline{AH}$ at $Q$ and $P$, respectively. What is $PQ$? [asy]draw((0,0)--(7,0)); draw((0,0)--(33/7,7.66651)); draw((33/7,7.66651)--(7,0)); draw((11/5,7*7.66651/15)--(7,0)); draw((63/17,0)--(33/7,7.66651)); draw((0,0)--(45/7,7.66651/4)); dot((0,0)); label("A",(0,0),SW); dot((7,0)); label("B",(7,0),SE); dot((33/7,7.66651)); label("C",(33/7,7.66651),N); dot((11/5,7*7.66651/15)); label("D",(11/5+.2,7*7.66651/15-.25),S); dot((63/17,0)); label("E",(63/17,0),NE); dot((45/7,7.66651/4)); label("H",(44/7,7.66651/4),NW); dot((27/7,3*7.66651/20)); label("P",(27/7,3*7.66651/20),NW); dot((5,7*7.66651/36)); label("Q",(5,7*7.66651/36),N); label("9",(33/14,7.66651/2),NW); label("8",(41/7,7.66651/2),NE); label("7",(3.5,0),S);[/asy] $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ \frac{5}{8}\sqrt{3} \qquad \textbf{(C)}\ \frac{4}{5}\sqrt{2} \qquad \textbf{(D)}\ \frac{8}{15}\sqrt{5} \qquad \textbf{(E)}\ \frac{6}{5}$

Let $ABCD$ an inscribed quadrilateral and $r$ and $s$ the reflections of the straight line through $A$ and $B$ over the inner angle bisectors of angles $\angle{CAD}$ and $\angle{CBD}$, respectively. Let $P$ the point of intersection of $r$ and $s$ and let $O$ the circumcentre of $ABCD$. Prove that $OP \perp CD$.

Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be the points on the sides $AD$ and $BC$ respectively such that $\frac{AP}{PD}=\frac{BQ}{QC}=\frac{AB}{CD}$. Prove that the line $PQ$ forms equal angles with the lines $AB$ and $CD$.

A convex quadrilateral intersects a circle at points $A_1,A_2,B_1,B_2,C_1,C_2,D_1,$ and $D_2$. (Note that for some letter $N$, points $N_1$ and $N_2$ are on one side of the quadrilateral. Also, the points lie in that specific order on the circle.) Prove that if $A_1B_2=B_1C_2=C_1D_2= D_1A_2$, then quadrilateral formed by these four segments is cyclic.

In a convex pentagon $ABCDE$ the following conditions hold : $AB \parallel CD$, $BC \parallel DE$, and $\angle BAE = \angle AED$. Prove that $AB + BC = CD + DE$ [i]Proposed by Anton Trygub[/i]

The point $P_1, P_2,\cdots ,P_{2018} $ is placed inside or on the boundary of a given regular pentagon. Find all placement methods are made so that $$S=\sum_{1\leq i<j\leq 2018}|P_iP_j| ^2$$takes the maximum value.

Consider a convex qudrilateral $ABCD$ and $M\in (AB),\ N\in (CD)$ such that $\frac{AM}{BM}=\frac{DN}{CN}=k$. Prove that $BC\parallel AD$ if and only if \[MN=\frac{1}{k+1} AD+\frac{k}{k+1} BC\] [i]***[/i]

Two circles of equal radius intersect at two distinct points $A$ and $B$. Let their radii $r$ and their midpoints respectively be $O_1$ and $O_2$. Find the greatest possible value of the area of the rectangle $O_1AO_2B$.

Let $ABC$ be a triangle and $\omega$ its incircle. $\omega$ touches $AC,AB$ at $E,F$, respectively. Let $P$ be a point on $EF$. Let $\omega_1=(BFP), \omega_2=(CEP)$. The parallel line through $P$ to $BC$ intersects $\omega_1,\omega_2$ at $X,Y$, respectively. Show that $BX=CY$.

We consider the quadrilateral $ABCD$, with $AD = CD = CB$ and $AB \parallel CD$. Points $E$ and $F$ belong to the segments $CD$ and $CB$ so that angles $\angle ADE = \angle AEF$. Prove that: a) $4CF \le CB$ , b) if $4CF = CB$, then $AE$ is the bisector of the angle $\angle DAF$.

Let $M$ and $N$ be points on the side $BC$ of a triangle $ABC$ such that $BM = CN$ ($M$ lies between $B$ and $N$). Points $P$ and $Q$ lie on $AN$ and $AM$ respectively, so that $\angle PMC =\angle MAB$ and $\angle QNB = \angle NAC$. Prove that $\angle QBC = \angle PCB$.

Triangle $ABC$ has $\angle{A}=90^\circ$, $\angle{C}=30^\circ$, and $AC=12$. Let the circumcircle of this triangle be $W$. Define $D$ to be the point on arc $BC$ not containing $A$ so that $\angle{CAD}=60^\circ$. Define points $E$ and $F$ to be the foots of the perpendiculars from $D$ to lines $AB$ and $AC$, respectively. Let $J$ be the intersection of line $EF$ with $W$, where $J$ is on the minor arc $AC$. The line $DF$ intersects $W$ at $H$ other than $D$. The area of the triangle $FHJ$ is in the form $\frac{a}{b}(\sqrt{c}-\sqrt{d})$ for positive integers $a,b,c,d,$ where $a,b$ are relatively prime, and the sum of $a,b,c,d$ is minimal. Find $a+b+c+d$.

Let $ABC$ be a triangle with $AB<AC<BC$.Let $O$ be the center of it's circumcircle and $D$ be the center of minor arc $\overarc{AB}$.Line $AD$ intersects $BC$ at $E$ and the circumcircle of $BDE$ intersects $AB$ at $Z$ ,($Z\not=B$).The circumcircle of $ADZ$ intersects $AC$ at $H$ ,($H\not=A$),prove that $BE=AH$.

The bisectors $AA_1, CC_1$ of a triangle $ABC$ with $\angle B = 60^{\circ}$ meet at point $I$. The circumcircles of triangles $ABC, A_1IC_1$ meet at point $P$. Prove that the line $PI$ bisects the side $AC$.

Inside the segment $AC$, an arbitrary point $B$ is selected and circles with diameters $AB$ and $BC$ are constructed. Points $M$ and $L$ are chosen on the circles (in one half-plane with respect to $AC$), respectively, so that $\angle MBA = \angle LBC$. Points $K$ and $F$ are marked, respectively, on rays $BM$ and $BL$ so that $BK = BC$ and $BF = AB$. Prove that points $M, K, F$ and $L$ lie on the same circle.